An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.

With G = GL(n, C), let XΓG be the G-character variety of a given finitely presented group Γ, and let X irr Γ G ⊂ XΓG be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for E-polynomials of XΓG and the one for X irr Γ G, generalizing a formula of Mozgovoy-Reineke [MR]. The proof uses a natural stratification of XΓG coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products; we also adapt it to the so-called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the E-polynomials, and Euler characteristics, of the irreducible stratum of GL(n, C)-character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of n.